In this paper, we show that near horizon analysis not only applies to AdS spacetime, but also to Lifshitz and Rindler geometry. The pole-skipping points are related to the hydrodynamic dispersion relations in Lifshitz geometry. In section , we calculate the lower half-plane pole-skipping points of tensor , Maxwell vector and Maxwell scalar mode in the background of the anisotropic system near Lifshitz points. The frequencies of these special points are located at negative integer (imaginary) Matsubara frequencies \(\mathfrak {w}_{n}:=\frac{\omega _n}{2\pi T}=-in\)  \((n=1,2,3\dots )\) . We study the complex hydrodynamic analyses and the analytic properties of the dispersion relations have been treated as Puiseux series in complex momentum [1]}. We plot the hydrodynamic dispersion relations in terms of dimensionless variables \(\frac{\omega }{2\pi T}\)  and \(\frac{\vert k\vert }{2\pi T}\) . We find that the hydrodynamic dispersion relations pass through pole-skipping points \((\mathfrak {w}_{n}, \vert \mathfrak {k}_{n}\vert )\)  along the direction of translation invariance at very small momentum and frequency. In section , we consider the axion field in the Lifshitz black hole with hyperscaling violating factor. The locations of lower half-plane pole-skipping points are the same at negative integer (imaginary) Matsubara frequencies. The hydrodynamic dispersion relation also fits the pole-skipping points well in the small \(k\)  limit in this section. In section , we calculate the pole-skipping points of scalar field in the background of AdS\(_2\times \mathbb {R}^{d-1}\) . The results from the near horizon analysis of the bulk equation are the same as the outcomes from the Green' function. We all know that the Green's function depends on the boundary conditions. For instance, choosing standard quantisation makes the Green's function different from choosing alternative quantisation for a scalar. But for the pole-skipping points, choosing different quantization will not affect their positions[2]}. We also verify the correctness of this conclusion in this section. In section , we obtain the “special points” of sound mode and scalar field in Rindler horizon. There is no boundary condition in Rindler geometry, so it still not enough to ensure a “pole-skipping” in the corresponding Green's function. But it is interesting that we can obtain “special points” from near horizon analysis which means there are two incoming solutions near the horizon which are related to the nonunique of the Green's function from a holographic point of view.
